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<a href="example7_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">/**</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment"> * @page example7 Example 7 : Unconditionally stable scheme for mechano-chemistry</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment"> * \dontinclude mechanoChemStab/userFunctions.cc</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment"> * As in previous examples, we include the header file declaring the required user functions.</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment"> * \line userFunction</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment"> * Then, we define initial/boundary conditions.</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment"> * &lt;b&gt; The \c boundaryConditions function &lt;/b&gt;</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment"> * We apply boundary conditions so that the material is free to move in tangential directions except on z=1, where we have the free-surface condition.  </span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="comment"> * \skip boundary</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="comment"> * &lt;b&gt; The \c cmuinit function and the \c uinit function &lt;/b&gt;</span></div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="comment"> * The initial conditions for the scalar and the vector fields are defined for this example problem as following.  </span></div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="comment"> * &lt;b&gt; The \c defineParameters function &lt;/b&gt;</span></div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;<span class="comment"> * Here, we define the mesh by setting the number of elements in each direction, e.g. a 64x64x64 element mesh.</span></div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;<span class="comment"> * \skip user.N[0]</span></div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;<span class="comment"> * \until user.N[2]</span></div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;<span class="comment"> * We also define the dimensions of the domain, e.g. a unit cube.</span></div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;<span class="comment"> * \skip user.L[0]</span></div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;<span class="comment"> * \until user.L[2]</span></div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;<span class="comment"> * We specify the number of vector and scalar solution and projection fields by adding the name of each field to</span></div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;<span class="comment"> * their respective vector. Here, we have one vector solution field (the displacement) and two scalar projection field</span></div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="comment"> * (the chemical composition and the chemical potential).</span></div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="comment"> * \skip &quot;c&quot;</span></div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;<span class="comment"> * \until &quot;displacement&quot;</span></div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;<span class="comment"> * We also specify the polynomial order of the basis splines and the global continuity.</span></div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;<span class="comment"> * \skip polyOrder</span></div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;<span class="comment"> * \until globalContinuity</span></div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;<span class="comment"> * We redirect the desired user function pointers to the \c boundaryConditions, \c cmuinit, and \c uinit functions that we</span></div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;<span class="comment"> * defined above.</span></div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;<span class="comment"> * \skip boundaryConditions</span></div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;<span class="comment"> * \until = uinit</span></div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;<span class="comment"> * We then define the timestep size.</span></div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;<span class="comment"> * \skip 1.e-4</span></div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;<span class="comment"> * \line dtVal</span></div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;<span class="comment"> * Finally, we define various (25) material parameters that describe the mechano-chemistry.</span></div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;<span class="comment"> * \skip malloc</span></div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;<span class="comment"> * \until par_mat[24]</span></div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;<span class="comment"> * &lt;b&gt; The \c residual function &lt;/b&gt;</span></div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;<span class="comment"> * The residual function for an unconditionally stable second-order scheme for mechano-chemistry is used in this example.</span></div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;<span class="comment"> * We first declare \c &quot;eval_residual&quot; non-member function to be used in the member function, \c &quot;residual&quot;.</span></div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;<span class="comment"> * \until eval_residual</span></div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;<span class="comment"> * The definition of the \c eval_residual function is postponed until the end of the file as it is lengthy.</span></div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;<span class="comment"> * For a complex problem like this example, where  it is convenient to unfold &quot;c.val&quot; and &quot;u.val&quot;, &quot;c.grad&quot; and &quot;u.grad&quot;, and &quot;c.hess&quot; and &quot;u.hess&quot; and put them into a single array, ui[]. </span></div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;<span class="comment"> * \skip ui[]=</span></div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;<span class="comment"> * \until };</span></div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;<span class="comment"> * We do the same for previous solutions represented by &quot;.valP&quot;, &quot;.gradP&quot;, and &quot;.hessP&quot; as well as for the test functions &quot;w1&quot; and &quot;w2&quot; and produce arrays, u0[] and w[], respectively.</span></div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;<span class="comment"> * We then evaluate the residual vector at a given quadrature point (residual[]) using the declared function &quot;eval_residual&quot;.</span></div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;<span class="comment"> * \skip T residual[</span></div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;<span class="comment"> * \until eval</span></div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;<span class="comment"> * Finally, we multiply residual[] by test functions and form the integrand of the weak form at the given quadrature point. </span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;<span class="comment"> * \skip r=0</span></div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;<span class="comment"> * \until dV</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;<span class="comment"> * A snippet of the code</span></div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;<span class="comment"> * ==============================</span></div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;<span class="comment"> * \dontinclude mechanoChemStab/userFunctions.cc</span></div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;<span class="comment"> * \skip userFunctions.h</span></div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;<span class="comment"> * \until comp[63]</span></div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;<span class="comment"> */</span></div>
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